5.2 Selection Rules and Transition Probabilities

Selection rules and transition probabilities are key to understanding atomic spectra. They determine which transitions between energy levels are allowed and how likely they are to occur. This knowledge helps predict spectral line intensities and interpret experimental results.

These concepts are crucial for analyzing atomic structure and behavior. By applying selection rules and calculating transition probabilities, we can explain observed spectral patterns and gain insights into the electronic configurations of atoms and molecules.

Selection Rules for Electric Dipole Transitions

Conservation of Angular Momentum and Parity

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• Selection rules for electric dipole transitions are based on the conservation of angular momentum and parity
• The change in total angular momentum quantum number (ΔJ) must be 0, ±1, with the restriction that the transition J = 0 to J = 0 is forbidden
• The change in magnetic quantum number (ΔMJ) must be 0, ±1
• The change in orbital angular momentum quantum number (ΔL) must be ±1
• The change in spin quantum number (ΔS) must be 0
• The change in parity (Δπ) must be odd, meaning the initial and final states must have opposite parity (even to odd or odd to even)

Quantum Number Changes

• ΔJ = 0, ±1 (J = 0 to J = 0 forbidden)
  • Example: A transition from a state with J = 1 to a state with J = 2 is allowed
• ΔMJ = 0, ±1
  • Example: A transition from a state with MJ = -1 to a state with MJ = 0 is allowed
• ΔL = ±1
  • Example: A transition from an S orbital (L = 0) to a P orbital (L = 1) is allowed
• ΔS = 0
  • Example: A transition between two states with the same spin multiplicity (singlet to singlet or triplet to triplet) is allowed
• Δπ = odd
  • Example: A transition from a state with even parity to a state with odd parity is allowed

Transition Probabilities and Spectral Line Intensities

Transition Probability and Line Intensity

• Transition probability is a measure of the likelihood of a specific transition occurring between two energy levels in an atom or molecule
• The transition probability is proportional to the square of the matrix element of the electric dipole moment operator between the initial and final states
• The intensity of a spectral line is directly proportional to the transition probability and the population of the initial state
• Higher transition probabilities result in stronger spectral lines, while lower transition probabilities lead to weaker spectral lines
• The Einstein coefficients (A and B) are used to describe the transition probabilities for spontaneous emission, stimulated emission, and absorption

Factors Influencing Spectral Line Intensities

• The population of the initial state can be determined using the Boltzmann distribution, which depends on the temperature and the energy difference between the initial state and the ground state
• Factors such as the degeneracy of the energy levels and the statistical weights of the states also influence the relative intensities of spectral lines
• The relative intensities of spectral lines can be compared by taking the ratio of their transition probability-population products
• Example: A transition with a higher transition probability and a larger population in the initial state will result in a more intense spectral line compared to a transition with a lower transition probability and a smaller initial state population

Allowed vs Forbidden Transitions

Allowed Transitions

• Allowed transitions are those that satisfy all the selection rules for electric dipole transitions
• Example: A transition from a 2P state to a 1S state (ΔL = -1, ΔJ = -1, ΔS = 0, Δπ = odd) is an allowed transition
• Allowed transitions typically result in strong spectral lines and have higher transition probabilities

Forbidden Transitions

• Forbidden transitions are those that violate one or more of the selection rules
• Transitions that violate the ΔJ rule (J = 0 to J = 0) are strictly forbidden
• Transitions that violate the ΔL, ΔS, or parity rules are considered forbidden but may still occur with lower probability through other mechanisms (e.g., magnetic dipole or electric quadrupole transitions)
• Example: A transition from a 1S state to another 1S state (ΔL = 0, ΔJ = 0) is forbidden and will not result in a spectral line
• Forbidden transitions have lower transition probabilities and may result in weak or absent spectral lines

Calculating Spectral Line Intensities

Relative Intensities

• The relative intensity of a spectral line is proportional to the product of the transition probability and the population of the initial state
• The transition probability can be calculated using the matrix element of the electric dipole moment operator between the initial and final states
• Example: If two transitions have the same initial state population but one has a transition probability twice that of the other, the spectral line corresponding to the higher transition probability will be twice as intense

Boltzmann Distribution and Temperature Dependence

• The population of the initial state can be determined using the Boltzmann distribution, which depends on the temperature and the energy difference between the initial state and the ground state
• The Boltzmann distribution is given by: $\frac{N_i}{N} = \frac{g_i e^{-E_i/kT}}{\sum_j g_j e^{-E_j/kT}}$, where $N_i$ is the population of state $i$, $N$ is the total population, $g_i$ is the degeneracy of state $i$, $E_i$ is the energy of state $i$, $k$ is the Boltzmann constant, and $T$ is the temperature
• At higher temperatures, the population of excited states increases, leading to changes in the relative intensities of spectral lines
• Example: In a high-temperature gas, transitions from higher excited states may become more intense due to the increased population of these states according to the Boltzmann distribution